These two lenses share the same focal lengths and almost the same apertures, but use completely different optical formulae to get there. They also use completely different mechanisms for moving the focus elements of the lenses. One also has IS (image stabilization) and the other one does not.

EF-S 18-55mm f/3.5-5.6 III

• Uses 11 elements in 9 groups, including one aspherical element in the rear group, and has 6 circular aperture blades.

• Has a micromotor AF motor that is the lowest type of AF motor in the Canon system. The few lenses in Canon's lineup with micromotor AF are generally slower and louder when focusing than most other AF types.
• No manual focus when AF is switched on (trying to manually focus when the AF/MF switch is set to 'AF' can damage the AF system), but does provide a direct mechanical connection between the focusing ring and the focusing elements when the AF/MF switch is set to 'MF'.
• No continuous AF when recording video.
• When the lens is zoomed in or out or the focus position changes, the front of the lens rotates. This affects the usability of certain types of filters such as polarizers or graduated neutral density filters.

EF-S 18-55mm f/4-5.6 IS STM

• Uses 12 elements in 10 groups, including an improved aspherical element in the rear group, and has 7 circular aperture blades.
• Has an STM (Stepping Motor) focusing motor that is optimized for quiet, smooth focusing appropriate for shooting video as well as stills. STM is not quite as fast as the top-of-the line USM AF motors found in more expensive Canon "L" series lenses, but they are generally smoother in transitioning from one focus distance to another. This is an advantage for shooting video using autofocus, but gives up ground when shooting stills of action or sports.
• STM technology does not allow a direct mechanical connection between the focusing ring and the lens' focus elements. Instead, moving the focusing ring sends an electronic signal to the camera which sends an instruction to the lens to move the focusing element using the STM. This 'focus-by-wire' system requires the lens to be attached to a powered on camera with active metering (such as when the shutter is half-pressed and the metering information shows in the viewfinder) in order to move the lens' focusing elements.
• Continuous AF when recording video with Canon bodies with Hybrid or Dual Pixel CMOS AF.
• Additionally, the EF-S 18-55mm f/4-5.6 IS STM includes optical Image Stabilization that counteracts small camera movements. This allows the photographer to use slower shutter speeds when hand holding the camera before noticeable blur from camera movement will affect an image. It does nothing to reduce motion blur due to the movement of anything other than the camera, such as your subject.
• The front of the lens does not rotate when the lens is zoomed in or out or when the focus position changes. This affects the usability of certain types of filters such as polarizers or graduated neutral density filters.

Conclusion

Other than the obvious differences in autofocus technology and IS (one has it, the other doesn't), there aren't lot of differences between these two lenses.

The newer EF-S 18-55mm f/4-5.6 IS STM introduced in 2017 is slightly better optically than the older 'III' introduced in 2011. The center of the frame is about equal for both lenses, but the further one goes from the center towards the edges the larger difference there is between the two, particularly at the longer focal lengths. Keep in mind that both of these lenses are lower end consumer grade lenses. Neither is going to be mistaken for a higher end zoom or even a consumer grade prime lens.

In exchange for a 1/3 stop slower maximum aperture from 18mm to 39mm where it finally catches up, the newer 'f/4-5.6 IS STM' is about 1/3 inch shorter when fully retracted than the older 'f/3.5-5.6 III'. At maximum extension both lenses are about the same length. The newer 'f/4-5.6 IS STM' weighs 0.7 ounces/20 grams more than the older 'f/3.5-5.6 III'.

Unless there is a substantial price difference between the two I would tend to recommend the newer EF-S 18-55mm f/4-5.6 IS STM over the EF-S 18-55mm f/3.5-5.6 III.

The only other factor that might change that is if you plan to do some "freelensing". The ability to change the focus position when the lens is not powered might come in handy there in certain situations. If you're reversing the lens for macro use the lens can be set at the minimum focus distance before removing it from the camera and you're probably not going to need to change the focus position while doing macro work (you usually focus by moving the camera towards/away from the subject to get the focus at the closest possible point to maximize the image size of the subject).